de Rham-Witt cohomology

Motivation: take $X\in {\mathsf{sm}}{\mathsf{Alg}}{\mathsf{Var}}_{/ {k}} $, and define ${ {H}^{\scriptscriptstyle \bullet}} _\mathrm{dR}(X_{/ {k}} ) \coloneqq { {{\mathbb{H}}}^{\scriptscriptstyle \bullet}} ( { {\Omega}^{\scriptscriptstyle \bullet}} _{X/k})$ using hypercohomology. Over ${\mathbb{C}}$ this will be isomorphic to singular cohomology and take values in ${\mathsf{k}{\hbox{-}}\mathsf{Mod}}$, but if $\operatorname{ch}k = p$ then the cohomology is entirely $p{\hbox{-}}$torsion. So Grothendieck/Berthelot define crystalline cohomology which takes values in ${\mathsf{W(k)}{\hbox{-}}\mathsf{Mod}}$, the $p{\hbox{-}}$typical Witt vectors. Originally this was defined in terms of the structure sheaf of the crystalline topos of $X$, but a more modern definition realizes it as ${ {H}^{\scriptscriptstyle \bullet}} _{\mathrm{crys}}(X_{/ {k}} ) = { {{\mathbb{H}}}^{\scriptscriptstyle \bullet}} ( { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}})$, the hypercohomology of the de Rham-Witt complex. This is a lift of algebraic de Rham in the following sense: $\cofib(p { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}} \to { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}}) \simeq { {\Omega}^{\scriptscriptstyle \bullet}} _{X/k}$ is a quasi-isomorphism of cochain complexes of sheaves.